Topological phases of matter have attracted much interest in recent years. Topological superconductors are of particular interest because they may host Majorana bound states [1]. Josephson junctions have been proposed as probes of topological superconductivity, and possible signatures of such Majorana bound states in topological Josephson junctions have indeed been observed [2,3,4]. However, important aspects related to the effect of the environment on the properties of the junction are still not fully understood. The aim of the thesis is to make progress in the understanding of quantum transport in voltage-biased topological Josephson junctions in the presence of an electromagnetic environment.

There is currently an active search for new phases of matter that admit topologically protected edge states. A promising route to realize them consists in combining conventional materials into appropriate heterostructures. Multiterminal Josephson junctions between conventional superconductors may be considered as topological materials themselves. As an example, 4-terminal junctions can accommodate topologically protected zero-energy bound states, which form so-called Weyl singularities. Their existence may be revealed through a quantized transconductance, like in the quantum Hall effect, but without magnetic field. The aim of the project will be to explore further this recent idea by investigating theoretically the robustness of this prediction in the presence of local Coulomb repulsion within the junction. In particular, the fate of Weyl singularities will be analyzed within an actual quantum-dot model for the junction.